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IEEE Information Theory Workshop (ITW)

An upper bound on the average probability of undetected error for the ensemble of all iterated codes

T. Nishijima

Published Jan 20, 2003• 1 citation
Publisher

Abstract

An upper bound on the average probability of undetected error for the ensemble of all iterated codes is given. It is assumed that the codes are binary linear block codes, and the channel is the binary symmetric channel.
DOI
10.1109/itcom.1999.781452

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By utilizing the method to get the upper bound on the average probability of undetected error for all binary linear systematic block codes, we newly get each upper bound on the average probability of undetected error for the ensembles of all binary product codes and all binary expansions of concatenated codes which include the constructive algebraic codes satisfying Shannon's channel coding theorem or being the asymptotically good. By comparing those upper bounds, we can clearly show that the average capability of concatenation structure is stronger than that of product codes, although the bound for concatenated codes is below that for the ensemble of all binary linear systematic block codes.
By utilizing the method to get the upper bound on the average probability of undetected error for all binary linear systematic block codes, we newly get each upper bound on the average probability of undetected error for the ensembles of all binary product codes and all binary expansions of concatenated codes which include the constructive algebraic codes satisfying Shannon's channel coding theorem or being the asymptotically good. By comparing those upper bounds, we can clearly show that the average capability of concatenation structure is stronger than that of product codes, although the bound for concatenated codes is below that for the ensemble of all binary linear systematic block codes.
The problem of computing the probability of undetected error is considered for linear block codes used for error detection. The recent literature is first reviewed and several results are extended. It is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual. Using the dual code formulation, the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes. Several bounds on the undetected error probability are then presented. We conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained. An efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results.